3.5.61 \(\int \frac {1}{x (-2+x^2)^{5/2}} \, dx\) [461]

Optimal. Leaf size=52 \[ -\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]

[Out]

-1/6/(x^2-2)^(3/2)+1/8*arctan(1/2*(x^2-2)^(1/2)*2^(1/2))*2^(1/2)+1/4/(x^2-2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 53, 65, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {x^2-2}}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {1}{4 \sqrt {x^2-2}}-\frac {1}{6 \left (x^2-2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(-2 + x^2)^(5/2)),x]

[Out]

-1/6*1/(-2 + x^2)^(3/2) + 1/(4*Sqrt[-2 + x^2]) + ArcTan[Sqrt[-2 + x^2]/Sqrt[2]]/(4*Sqrt[2])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (-2+x^2\right )^{5/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(-2+x)^{5/2} x} \, dx,x,x^2\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{(-2+x)^{3/2} x} \, dx,x,x^2\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {-2+x} x} \, dx,x,x^2\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-2+x^2}\right )\\ &=-\frac {1}{6 \left (-2+x^2\right )^{3/2}}+\frac {1}{4 \sqrt {-2+x^2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 46, normalized size = 0.88 \begin {gather*} \frac {-8+3 x^2}{12 \left (-2+x^2\right )^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-2+x^2}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-2 + x^2)^(5/2)),x]

[Out]

(-8 + 3*x^2)/(12*(-2 + x^2)^(3/2)) + ArcTan[Sqrt[-2 + x^2]/Sqrt[2]]/(4*Sqrt[2])

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Maple [A]
time = 0.11, size = 37, normalized size = 0.71

method	result	size
risch	\(\frac {3 x^{2}-8}{12 \left (x^{2}-2\right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right )}{8}\)	\(35\)
default	\(-\frac {1}{6 \left (x^{2}-2\right )^{\frac {3}{2}}}+\frac {1}{4 \sqrt {x^{2}-2}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right )}{8}\)	\(37\)
trager	\(\frac {3 x^{2}-8}{12 \left (x^{2}-2\right )^{\frac {3}{2}}}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\sqrt {x^{2}-2}-\RootOf \left (\textit {\_Z}^{2}+2\right )}{x}\right )}{8}\)	\(47\)
meijerg	\(\frac {\sqrt {2}\, \left (-\mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )\right )^{\frac {5}{2}} \left (\frac {3 \left (\frac {8}{3}-3 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-6 x^{2}+16\right )}{8 \left (-\frac {x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {x^{2}}{2}+1}}{2}\right )}{2}\right )}{12 \sqrt {\pi }\, \mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )^{\frac {5}{2}}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^2-2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/6/(x^2-2)^(3/2)+1/4/(x^2-2)^(1/2)-1/8*2^(1/2)*arctan(1/(x^2-2)^(1/2)*2^(1/2))

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Maxima [A]
time = 2.46, size = 33, normalized size = 0.63 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \arcsin \left (\frac {\sqrt {2}}{{\left | x \right |}}\right ) + \frac {1}{4 \, \sqrt {x^{2} - 2}} - \frac {1}{6 \, {\left (x^{2} - 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-2)^(5/2),x, algorithm="maxima")

[Out]

-1/8*sqrt(2)*arcsin(sqrt(2)/abs(x)) + 1/4/sqrt(x^2 - 2) - 1/6/(x^2 - 2)^(3/2)

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Fricas [A]
time = 0.59, size = 65, normalized size = 1.25 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{4} - 4 \, x^{2} + 4\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} x + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - 2}\right ) + {\left (3 \, x^{2} - 8\right )} \sqrt {x^{2} - 2}}{12 \, {\left (x^{4} - 4 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(3*sqrt(2)*(x^4 - 4*x^2 + 4)*arctan(-1/2*sqrt(2)*x + 1/2*sqrt(2)*sqrt(x^2 - 2)) + (3*x^2 - 8)*sqrt(x^2 -
2))/(x^4 - 4*x^2 + 4)

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Sympy [C] Result contains complex when optimal does not.
time = 2.18, size = 984, normalized size = 18.92 \begin {gather*} \begin {cases} \frac {6 i x^{4} \log {\left (x \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {3 i x^{4} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {6 x^{4} \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {6 \sqrt {2} x^{2} \sqrt {x^{2} - 2}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {24 i x^{2} \log {\left (x \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 i x^{2} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {24 x^{2} \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {16 \sqrt {2} \sqrt {x^{2} - 2}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {24 i \log {\left (x \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 i \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {24 \operatorname {asin}{\left (\frac {\sqrt {2}}{x} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} & \text {for}\: \left |{x^{2}}\right | > 2 \\- \frac {3 i x^{4} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {6 i x^{4} \log {\left (\sqrt {1 - \frac {x^{2}}{2}} + 1 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {3 \pi x^{4}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {3 i x^{4} \log {\left (2 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {6 \sqrt {2} i x^{2} \sqrt {2 - x^{2}}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 i x^{2} \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {24 i x^{2} \log {\left (\sqrt {1 - \frac {x^{2}}{2}} + 1 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 \pi x^{2}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 i x^{2} \log {\left (2 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {16 \sqrt {2} i \sqrt {2 - x^{2}}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 i \log {\left (x^{2} \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {24 i \log {\left (\sqrt {1 - \frac {x^{2}}{2}} + 1 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} - \frac {12 \pi }{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} + \frac {12 i \log {\left (2 \right )}}{24 \sqrt {2} x^{4} - 96 \sqrt {2} x^{2} + 96 \sqrt {2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**2-2)**(5/2),x)

[Out]

Piecewise((6*I*x**4*log(x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 3*I*x**4*log(x**2)/(24*sqrt(2)*x
**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 6*x**4*asin(sqrt(2)/x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2))
+ 6*sqrt(2)*x**2*sqrt(x**2 - 2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 24*I*x**2*log(x)/(24*sqrt(2
)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 12*I*x**2*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2))
+ 24*x**2*asin(sqrt(2)/x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 16*sqrt(2)*sqrt(x**2 - 2)/(24*sqr
t(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 24*I*log(x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 12*
I*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 24*asin(sqrt(2)/x)/(24*sqrt(2)*x**4 - 96*sqrt(2
)*x**2 + 96*sqrt(2)), Abs(x**2) > 2), (-3*I*x**4*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) +
6*I*x**4*log(sqrt(1 - x**2/2) + 1)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 3*pi*x**4/(24*sqrt(2)*x*
*4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 3*I*x**4*log(2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 6*sqrt
(2)*I*x**2*sqrt(2 - x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 12*I*x**2*log(x**2)/(24*sqrt(2)*x
**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 24*I*x**2*log(sqrt(1 - x**2/2) + 1)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 +
96*sqrt(2)) + 12*pi*x**2/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 12*I*x**2*log(2)/(24*sqrt(2)*x**4
- 96*sqrt(2)*x**2 + 96*sqrt(2)) - 16*sqrt(2)*I*sqrt(2 - x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)
) - 12*I*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 24*I*log(sqrt(1 - x**2/2) + 1)/(24*sqrt(
2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 12*pi/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 12*I*log(2)
/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)), True))

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Giac [A]
time = 0.75, size = 35, normalized size = 0.67 \begin {gather*} \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - 2}\right ) + \frac {3 \, x^{2} - 8}{12 \, {\left (x^{2} - 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-2)^(5/2),x, algorithm="giac")

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x^2 - 2)) + 1/12*(3*x^2 - 8)/(x^2 - 2)^(3/2)

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Mupad [B]
time = 0.47, size = 34, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x^2-2}}{2}\right )}{8}+\frac {\frac {x^2}{4}-\frac {2}{3}}{{\left (x^2-2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 - 2)^(5/2)),x)

[Out]

(2^(1/2)*atan((2^(1/2)*(x^2 - 2)^(1/2))/2))/8 + (x^2/4 - 2/3)/(x^2 - 2)^(3/2)

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